Learn English – The usage of “to try out” and “to try”

First of all, let's be clear that ain't is not commonly used in proper and correct speech. It is a word, it is technically correct, and it does take the meaning of "[to be] not", as you have surmised. However it is most often used in uneducated speech, so you must be sure to take everything with a grain of salt when you see it used. I certainly wouldn't suggest you begin using it in formal contexts.

Let's begin with your first assertion:

I ain't an idiot

expands to

I am not an idiot

Which, in my understanding means "Well, I am declaring myself an idiot"

Ain't does indeed expand to am not here, but it certainly doesn't mean that the person thinks they are an idiot. They are quite clearly saying that they are not an idiot (probably in response to someone who has said they are one). I am not [x] is always a claim that a person is not something.

It just ain't done good

expands to

It just have not done good

Which, in my understanding means "An action didn't generated good results of if"

Now we return to my original point that ain't is often used in uneducated speech. The formal way to say this would be "It just is not done well." This means that [x] action (or the result of said action) has not been performed properly. For example, a teacher could be looking at a student's painting in an art class, and comment that the painting was not done (ie. not painted) well. But ain't doesn't expand to have not here, as you suggest; it expands to is not (simply another form of [to be] not).

But I ain't marchin' anymore

expands to

But I am not marching anymore

Which, in my understanding means "I am saying, that I will stop marching"

Your expansion to am not is correct, and you've almost got the interpretation down; what this means is I am no longer marching. I have not been marching for some period time up until this point, and I will continue not to march. The distinction from your interpretation is that they are not saying "I will stop marching from this point forward." They have already stopped marching.

Hey, ain't gonna cry no more today

expands to

Hey, I am not going to cry no more today

This one is a bit weird, but in my understanding means "I am declaring my intentions of crying no more that I already did, today"

Again we return to the issue of uneducated speech; the ain't is actually completely straightforward in this example, it does indeed expand to am not. However you have a different problem which is displayed in uneducated speech; that of the word no used when it ought to be any. This could have been correctly written I ain't going to cry any more today (expanding to I am not going to cry any more today). This means that at some point today they have cried a certain amount, and intend to not cry again for the rest of the day. They will not cry any more (or any longer).

I ain't no quitter

expands to

I am not no quitter

Which, in my understanding means that "I am a quitter", since be first negation negates the second...

This example has the same problem as the last one; the ain't is straight forward but the no is incorrect. It should read I ain't a quitter (expanded to I am not a quitter). Which, similar to your first example where the person professes to not be an idiot, means that the person denies any accusation that they might be a quitter.

So, overall, be careful with ain't. You've got a good handle on the expansion overall, but you have to watch out for the extra no/not negations you might find. Use context to help you as much as possible; rarely would someone claim to be a quitter or an idiot, so your best guess should be that it means the opposite :)

Checking something out can mean "taking a look at something" when used colloquially. So with the first three sentences, checking out does not necessarily mean that you're going to eat there.

If you say "I am going to check that restaurant out" it can both mean "I am going to read something about them, get myself informed" or "I'm going there to try their food". However it usually means the latter, but that's not a rule.

Try out / try is fairly interchangeable in your example sentences, although it sounds better without the "out" to my ears.

Situation 1:

You would say "Have you checked your bag?", not "Have you checked out your bag?".

And yes, that's another way of saying "Have you looked in your bag?"

Situation 2:

"Could you check your inbox?" is colloquial but it would work. As would "Could you [please] check your email [as soon as possible]?". The word "mailbox" however can't really be used when talking about an email as that word is commonly used to refer to a physical mailbox (The one in front of your house).